Advertisement

Archiv der Mathematik

, Volume 111, Issue 3, pp 313–327 | Cite as

On the bidomain problem with FitzHugh–Nagumo transport

  • Matthias Hieber
  • Jan Prüss
Article

Abstract

The bidomain problem with FitzHugh–Nagumo transport is studied in the \(L_p\!-\!L_q\)-framework. Reformulating the problem as a semilinear evolution equation, local well-posedness is proved in strong as well as in weak settings. We obtain solvability for initial data in the critical spaces of the problem. For dimension \(d\le 4\), by means of energy estimates and a recent result of Serrin type, global existence is shown. Finally, stability of spatially constant equilibria is investigated, to the result that the stability properties of such equilibria are the same as for the classical FitzHugh–Nagumo system in ODE’s. These properties of the bidomain equations are obtained combining recent results on the bidomain operator (Hieber and Prüss in Theory for the bidomain operator, submitted, 2018), on critical spaces for parabolic evolution equations (Prüss et al in J Differ Equ 264:2028–2074, 2018), and qualitative theory of evolution equations.

Keywords

Bidomain operator Maximal \(L_p\)-regularity FitzHugh–Nagumo transport Critical spaces Global existence Stability 

Mathematics Subject Classification

35K50 92C35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. Amann, Linear and Quasilinear Parabolic Problems I, Monographs in Mathematics, 89, Birkhäuser, 1995.CrossRefGoogle Scholar
  2. 2.
    L. Ambrosio, P. Colli Franzone, and G. Savaré, On the asymptotic behaviour of anisotropic energies arising in the cardiac bidomain model, Interfaces Free Bound. 2 (2000), 213–266.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Y. Bourgault, Y. Coudière, and C. Pierre, Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology, Nonlinear Anal. Real World Appl. 10 (2009), 458–482.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level, In: Evolution Equations, Semigroups and Functional Analysis, Progr. Nonlinear Differential Equations Appl., vol. 50, Birkhäuser, Basel, 2000.Google Scholar
  5. 5.
    P. Colli Franzone, L. Pavarino, and S. Scacchi, Mathematical Cardiac Electrophysiology, Springer, 2014.CrossRefzbMATHGoogle Scholar
  6. 6.
    Y. Giga and N. Kajiwara, On a resolvent estimate for bidomain operators and its applications, J. Math. Anal. Appl. 459 (2018), 528–555.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    M. Hieber and J. Prüss, \(L_q\)-Theory for the bidomain operator, submitted, 2018.Google Scholar
  8. 8.
    T. Hytönen, J. van Neerven, M. Veraar, and L. Weis, Analysis in Banach. Vol. I. Martingales and Littlewood-Paley theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics, 63, Springer, Cham, 2016.CrossRefzbMATHGoogle Scholar
  9. 9.
    J. Keener and J. Sneyd, Mathematical Physiology, Interdisciplinary Applied Mathematics, 8, Springer-Verlag, New York, 1998.Google Scholar
  10. 10.
    K. Kunisch and M. Wagner, Optimal control of the bidomain system (IV): corrected proofs of the stability and regularity theorems, arXiv:1409.6904v2.
  11. 11.
    Y. Mori and H. Matano, Stability of front solutions of the bidomain equation, Comm. Pure Appl. Math. 69 (2016), 2364–2426.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    M. Pennacchio, G. Savaré, and P. Colli Franzone, Multiscale modeling for the bioelectric activity of the heart, SIAM J. Math. Anal. 37 (2005), 1333–1370.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, 105, Birkhäuser, 2016.Google Scholar
  14. 14.
    J. Prüss, G. Simonett, and M. Wilke, Critical spaces for quasilinear parabolic evolution equations and applications, J. Differential Equations. 264 (2018), 2028–2074.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    M. Veneroni, Reaction-diffusion systems for the macroscopic bidomain model of the cardiac electric field, Nonlinear Anal. Real World Appl. 10 (2009), 849–868.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Institut für MathematikMartin-Luther-Universität Halle-WittenbergHalleGermany

Personalised recommendations